reserve X for non empty set;
reserve e for set;
reserve x for Element of X;
reserve f,g for PartFunc of X,ExtREAL;
reserve S for SigmaField of X;
reserve F for Function of RAT,S;
reserve p,q for Rational;
reserve r for Real;
reserve n,m for Nat;
reserve A,B for Element of S;

theorem Th20:
for C being non empty set, f being PartFunc of C,ExtREAL, x being Element of C
  st x in dom f & max+(f).x = 0. holds max-(f).x = -(f.x)
proof
  let C be non empty set;
  let f be PartFunc of C,ExtREAL;
  let x be Element of C;
  assume that
A1: x in dom f and
A2: max+(f).x = 0.;
A3: x in dom(max+(f)) by A1,Def2;
A4: x in dom(max-(f)) by A1,Def3;
A5: max+(f).x = max(f.x,0.) by A3,Def2;
A6: max-(f).x = max(-(f.x),0.) by A4,Def3;
 f.x <= 0. by A2,A5,XXREAL_0:def 10;
  hence thesis by A6,XXREAL_0:def 10;
end;
